3 THE MODULUS FUNCTION ON A GROUP 257 a = 1 (because a > 0). This proves the first assertion. The second then follows (13.14.3). We deduce that if /is a locally /j-integrable function, then (14.3.4.2) (/•AO~=(A-1/)-M. (14.3.5) In particular, if G is unimodular> then for each ju-integrable function the functions yC?)/, §(s)f and / are /i-integrable, and we have .3.5.1) f f(sx) dtfd = ( (14.3 In particular, if A is any /i-integrable set in G, then (14.3.5.2) for all s e G. When G is infinite and compact (resp. infinite and discrete) and therefore unimodular (14.3.3), the normalized Haar measure on G is the unique Haar measure \JL on G for which n(G) = 1 (resp. n({e}) = 1). (14.3.6) Now let u be an automorphism of the topological group G. It is clear that the image U~"I(IJL) of a left Haar measure \JL on G (13.1.6) is another left Haar measure; hence (14.1.5) there exists a number a > 0, independent of the choice of ju, such that U~I(IJL) = aju. This number a is called the modulus of the automorphism uy and is denoted by modG(w) or mod(w). For every ^-integrable function/we have therefore f t f (14.3.6.1) f(u~ \x)) d^(x) = (mod u) | /(x and in particular, for any /i-integrable set A, (14.3.6.2) Mw(A)) = (mod u In particular, for each s e G, let is be the inner automorphism Then we have /71 = &(S)Y(S\ an^ therefore which proves that (14.3J) mod(fs) =1 • /i) = (50A-1) • (8(^) = (A^^A-1) • (A^)^) = A"1 - u,